Question: The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points $(x_1,y_1),$ $(x_2,y_2),$ $(x_3,y_3),$ and $(x_4,y_4).$  Find
\[x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.\]
To find $x_1 + x_2 + x_3 + x_4,$ we can try to find a quartic equation whose roots are $x_1,$ $x_2,$ $x_3,$ and $x_4.$  To this end, we substitute $y = (x + 1)^2$ into $x + 4 = (y - 3)^2,$ to get
\[x + 4 = ((x + 1)^2 - 3)^2.\]Expanding, we get $x^4 + 4x^3 - 9x = 0.$  By Vieta's formulas, $x_1 + x_2 + x_3 + x_4 = -4.$

Substituting $x = (y - 3)^2 - 4$ into $y = (x + 1)^2,$ we get
\[y = ((y - 3)^2 - 3)^2.\]Expanding, we get $y^4 - 12y^3 + 48y^2 - 73y + 36 = 0.$  By Vieta's formulas, $y_1 + y_2 + y_3 + y_4 = 12.$

Therefore, $x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4 = \boxed{8}.$